A318772 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.

1, 1, 1, 1, 1, 3, 1, 6, 1, 9, 1, 12, 1, 15, 9, 1, 18, 27, 1, 21, 54, 1, 24, 90, 1, 27, 135, 27, 1, 30, 189, 108, 1, 33, 252, 270, 1, 36, 324, 540, 1, 39, 405, 945, 81, 1, 42, 495, 1512, 405, 1, 45, 594, 2268, 1215, 1, 48, 702, 3240, 2835, 1, 51, 819, 4455, 5670, 243, 1, 54, 945, 5940, 10206, 1458

Offset

0,6

Comments

The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-right in center-justified triangle given in A013610 ((1+3*x)^n) and along a "third layer" skew diagonals pointing top-left in center-justified triangle given in A027465 ((3+x)^n), see links. (Note: First layer of skew diagonals in center-justified triangles of coefficients in expansions of (1+3*x)^n and (3+x)^n are given in A304236 and A304249 respectively.)

The coefficients in the expansion of 1/(1-x-3x^4) are given by the sequence generated by the row sums.

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.6580980673722..., when n approaches infinity.

References

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Links

G. C. Greubel, Rows n = 0..120 of the irregular triangle, flattened

Zagros Lalo, Third layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 3x)^n

Zagros Lalo, Third layer skew diagonals in center-justified triangle of coefficients in expansion of (3 + x)^n

Formula

T(n,k) = 3^k * (n - 3*k)!/ ((n - 4*k)! k!), where n >= 0 and 0 <= k <= floor(n/4).

Example

Triangle begins:
  1;
  1;
  1;
  1;
  1,  3;
  1,  6;
  1,  9;
  1, 12;
  1, 15,   9;
  1, 18,  27;
  1, 21,  54;
  1, 24,  90;
  1, 27, 135,   27;
  1, 30, 189,  108;
  1, 33, 252,  270;
  1, 36, 324,  540;
  1, 39, 405,  945,   81;
  1, 42, 495, 1512,  405;
  1, 45, 594, 2268, 1215;
  ...

Mathematica

T[n_, k_]:= T[n, k]= 3^k(n-3k)!/((n-4k)! k!); Table[T[n, k], {n, 0, 21}, {k, 0, Floor[n/4]} ]  // Flatten.
T[0, 0] = 1; T[n_, k_] := T[n, k] = If[n<0 || k<0, 0, T[n-1, k] + 3T[n-4, k-1]]; Table[T[n, k], {n, 0, 21}, {k, 0, Floor[n/4]}] // Flatten.

Prog

(Magma) [3^k*Binomial(n-3*k, k): k in [0..Floor(n/4)], n in [0..24]]; // G. C. Greubel, May 12 2021
(Sage) flatten([[3^k*binomial(n-3*k, k) for k in (0..n//4)] for n in (0..24)]) # G. C. Greubel, May 12 2021

Crossrefs

Row sums give A318774.

Cf. A013610, A027465.

Cf. A304236, A304249.

Cf. A317496, A317497.

Sequence in context: A109646 A199783 A329645 * A317496 A304236 A360654

Adjacent sequences: A318769 A318770 A318771 * A318773 A318774 A318775

Keyword

tabf,nonn,easy

Author

Zagros Lalo, Sep 04 2018

Status

approved